To solve this problem, we can use the fact that the sum of angles in any triangle is always $180^\circ$.

In triangle $\triangle MNL$, we have three angles:

1. $\angle M = 43^\circ$
2. $\angle N = (5x - 6)^\circ$
3. $\angle L = (3x - 9)^\circ$

The equation for the sum of angles in the triangle is: $43 + (5x - 6) + (3x - 9) = 180$

Step-by-Step Solution

1. Combine like terms: $43 + 5x - 6 + 3x - 9 = 180$ $8x + 28 = 180$

2. Subtract 28 from both sides: $8x = 152$

3. Divide by 8: $x = 19$

So, the value of $x$ is $\mathbf{19}$.

Would you like further details on this solution or have any questions?

Here are some related questions to explore:

1. What if the angles were given as different expressions?
2. How do you find each angle after knowing $x$?
3. Why is the sum of angles in a triangle always $180^\circ$?
4. Can you apply this method to other types of polygons?
5. How does this relate to other geometry concepts, such as exterior angles?

Tip: When solving equations with multiple terms, always combine like terms to simplify the equation first.